Consider the induced morphism of Weil restriction of scalars \(R_{L/k}(\psi _L):R_{L/k} Z^{\prime }_L \rightarrow R_{L/k} Z_L\) ; see [1]}. Let \(\Delta :Z\rightarrow R_{L/k} Z_L\)  be the diagonal morphism. Let \((W_j)_{j\in J}\)  be the connected components of \(Z\times _{\Delta , R_{L/k} Z_L} R_{L/k} Z^{\prime }_L\)  such that \(W_j(k)\ne \emptyset \)  and let \(f_j:W_j\rightarrow Z\)  be the natural morphism.
Since \(W_j\)  is normal connected and with a \(k\) -rational point, it is geometrically integral. Moreover, note that, by the defining properties of the Weil restriction of scalars, the finite set of covers \(\lbrace f_j:W_j\rightarrow Z\rbrace _{j\in J}\)  is an arithmetic refinement of \(Z\) .
